Riemannian Optimization Of Variational Quantum Algorithms

Quantum Natural Gradient (QNG) is widely considered the theoretically optimal optimizer for variational quantum circuits. However, QNG can become unstable as variational quantum circuits become deeper and noisier, yet the geometric origin of this instability remains poorly understood. We uncover a noise-induced Riemannian spike (NIRS): under local Markovian noise, the pullback metric can vanish exponentially faster than the Euclidean gradient, causing metric inversion to amplify the natural-gradient update exponentially with circuit depth. This reveals a previously overlooked failure mechanism in which noise can destabilize optimization through metric inversion, rather than through gradient suppression alone. In parallel, we introduce Dually Flat Stratified Quantum Natural Gradient (DFS-QNG), a structured Riemannian optimizer that exploits commuting parameter strata. By restricting each update to a commuting submanifold, DFS-QNG obtains exact blockwise geodesic updates while replacing costly global metric inversion with smaller blockwise systems. We further establish geodesic smoothness and convergence guarantees for the resulting Riemannian optimization framework. Experiments on noisy and noiseless VQE and QAOA circuits demonstrate improved stability and faster convergence than standard QNG and Adam, together with substantially reduced metric computation.